-
Dynamical Structure and Spectral Properties ofInput-Output
Networks
Ernest Liu and Aleh Tsyvinski∗
November 30, 2020
Abstract
We associate a dynamical system with input-output networks and
study its spectralproperties. Specifically, we develop a dynamic
production network model featuringadjustment costs of changing
inputs and thus gradual recovery from temporary TFPshocks. First,
we explicitly solve for the output and welfare effects of temporary
shocks.We show shocks to sectors that generate significant sales
through distant linkages to theconsumer are most damaging. Second,
we eigendecompose the input-output matrix andshow, because
higher-order linkages take longer to recover, fewer eigenvectors
are neededto represent the welfare impact of sectoral shocks in the
dynamic economy comparedto the Domar weights. Third, we analyze the
U.S. input-output structure and showthe welfare impact of temporary
shocks has a low-dimensional, 4-factor structure (outof 171
eigenvectors). Finally, we revisit the historical use of
input-output analysis intarget selection for bombing Nazi Germany
and Imperial Japan during WWII.
∗Liu: Princeton University; Tsyvinski: Yale University. We thank
Daron Acemoglu, Ruben Enikolopov,Benny Kleinman, Stephen Redding,
and Stefan Steinerberger for comments, and Yinshan Shang and
Alexan-der Zimin for research assistance.
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1 Introduction
The study of production networks recently became an active
research agenda in macroeco-nomics, reviving and developing the
classic analysis of Leontief. The most common settingis that of the
static general equilibrium environment subject to sectoral
shocks.
The recent theoretical advances in the analysis of networks and
graphs take an inherentlydynamical perspective. Spectral graph
theory (Chung (1997), Grigor’yan (2018), Spielman(2019)) associates
a dynamical system, typically a diffusion operator, with the static
networkand studies the eigenvectors and eigenvalues of the
Laplacian. The Leontief-inverse matrixin the input-output analysis
is the inverse of the Laplacian of the network and thus admitsa
parallel analysis. Similarly, a classical way to analyze a
non-negative matrix (Berman andPlemmons (1994)), which is the
input-output network, is by associating with it a continuoustime
dynamical system, or a flow, represented by a system of
differential equations.1 Thissystem has an explicit solution in
terms of matrix exponentials and hence the propertiesof solutions
are intimately related to the properties of the matrix (Colonius
and Kliemann(2014)). In particular, the eigenvalues and
eigenvectors of the input-output matrix deter-mine the evolution of
the dynamical system and thus can be used for the analysis of
shockpropagations. Our main methodological aim is to bring the
techniques of dynamical systemsand spectral graph theory to the
analysis of production networks.
In this paper, such a dynamical system arises from a dynamic
model of productionnetworks in which there are costs for changing
inputs of production. Following a temporarynegative shock,
adjustment costs lead to a gradual movement of the economy to the
steadystate. While the steady state of the economy coincides with
the static model, the transitionpath of the dynamic transmission of
shocks across sectors is in general different from thestandard
setting and the temporary shocks have lasting impact. Our primary
goal is tocharacterize the dynamic path of the propagation of
shocks through the input-output linkagesand the determinants of the
output trajectory and welfare.
Specifically, we extend the model of Acemoglu, Carvalho,
Ozdaglar, and Tahbaz-Salehi(2012) and Jones (2011, 2013) by
introducing a cost of upward change of inputs that dependson the
speed of adjustment. The standard model corresponds to the case of
zero adjustmentcosts and thus an immediate adjustment to the
temporary shocks. For positive adjustmentcosts, the standard model
is a steady state of our dynamic model. We show that analysisof the
determinants of the flow of output (that is, the solution to the
dynamical system ofdifferential equations) is closely related to
the properties of the static input-output matrix
1The behavior of the Markov chain on the input-output network is
a closely related dynamical system(Kemeny and Snell (1960)).
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as in the mathematical approaches described above. The analysis
of welfare impact oftemporary shocks also adds new considerations
in analyzing the temporal structure of theflow.
We first characterize the transition path of the sectoral
outputs and consumptions. Evenwhen the sectoral TFP recovery is
immediate, due to adjustment costs, the use of interme-diate goods
cannot jump instantaneously and only recovers gradually. The
gradual recoveryof the use of inputs then translates into the
gradual recovery of the output in sectors thatuse those inputs. As
the transition takes time, the speed of which is determined by the
mag-nitude of the adjustment costs, the path of adjustment has
non-trivial welfare implicationsfor a consumer who discounts the
future. We thus next characterize the impact of the sec-toral TFP
shock on consumer welfare. We show that the elasticity of welfare
to temporarynegative TFP shocks is proportional to the difference
between two terms. The first is thesectoral Domar weight that
captures the welfare impact of a permanent negative TFP shock.The
second term captures the effects of the slow recovery of the
production network. Thisterm has a similar form to the Domar weight
but where the power series of the subsequentround of effects on
production is discounted by the product of the consumer’s discount
rateand the adjustment cost parameter. Temporary shocks have
lasting impact on the outputand welfare precisely because
input-output linkages take time to recover. These
higher-orderlinkages are represented by the series of the powers of
input-output matrix. Importantly, thedifference between these two
terms disproportionately removes the lower-order rounds of
theproduction effects while keeping the tail entries unchanged.
For example, one can contrast two different networks. In a
horizontal network wherethere are no input-output linkages, the
adjustment costs are irrelevant, and a negative TFPshock has zero
impact on the economy once the TFP reverts. In a vertical economy,
wheresectors are ordered and each supplies only to the next one, in
contrast, temporary shockshave a lasting impact, and the damage is
more severe when the shocked sector is more distantfrom the final
consumer.
We then show that the welfare measure is similar to the concept
of alpha centrality in thenetwork literature with the difference
that the weights increase for the higher-order linkages.One can
view this also as a multi-scale representation of the economy. We
show that theimportance of the higher-order links is determined by
a parameter that is a combination ofthe speed of adjustment and the
discount rate of the consumer. This one-parameter family
ofeconomies then shows the importance of temporary shocks’ local
versus global effects. Thatis, the economy is represented at
different scales or the levels of coarseness spotlighting
therelative importance of the higher-order links and thus the
importance of the global versuslocal structures.
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Summarizing our first main result of the paper: in the presence
of adjustment frictions,we derive the explicit representation for
the output and welfare effects of the temporaryshocks.
Specifically, the shocks to sectors that generate significant sales
through distantlinkages to the consumer are disproportionately
damaging to the economy.
Our second set of results is the characterization of the main
driving forces of the welfareimpact of temporary shocks. Since we
have shown the importance of the higher-order pro-duction links,
this naturally leads us to analyze the spectral or
eigendecomposition of theinput-output matrix—its eigenvectors and
eigenvalues. The main reason for this is that oncethe matrix is
diagonalized, the powers of it, which represent the higher order
productionlinks, take a particularly simple form.
The concise decomposition of the welfare effects of shocks as a
combination of the eigen-vectors and the power series of the
eigenvalues is our second main theoretical result. Specif-ically,
consider a temporary shock vector that is itself an eigenvector.
The impact of theshock along the entire transition path becomes a
continuously decayed version of the initialshock, with rate of
decay governed by the eigenvalue. An important corollary of this
logicshows a marked contrast with the eigendecomposition of the
Domar weights, and thus withthe static economy. Because dynamic
adjustment costs significantly down-weight the directand initial
rounds of network effects, our model effectively up-weights the
higher powers inthe power series of eigenvalues, thereby
up-weighting the relative importance for the shockprofiles with
greater eigenvalues and down-weighting the shock profiles with the
lower eigen-values. This implies that fewer eigenvalues are needed
to represent the welfare impact ofthe shocks in a dynamic economy.
In other words, the dynamic economy may have a factorstructure
where the small set of factors can capture the importance of
temporary shocks. Incontrast, the Domar weights may be
significantly higher dimensional: because the Domarweights do not
discount the direct and initial rounds of network effects, even
eigenvectorswith small eigenvalues may have a sizable contribution
in explaining TFP shocks in thestatic model. In summary, sectoral
shocks may not have a low-dimensional representationin the static
model but may have one in our dynamic model. The concise
representation ofcomplex high dimensional systems via a few reduced
coordinates is also the primary goal ofthe well developed
literature on nonlinear dimensionality reduction using spectral
methods(e.g., Coifman, Kevrekidis, Lafon, Maggioni, and Nadler
(2008)).
Our third set of results is an empirical analysis of the
eigendecomposition of the U.S.input-output structure. We first show
that 95 percent of the welfare effect of temporarysectoral shocks
can be represented by only four eigenvectors. That is, the U.S.
input-outputnetwork has a very low- dimensional (4-factor)
structure. In contrast, for the Domar weightalmost all of the 171
eigenvectors are important, and, hence, the Domar weight is a
high-
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dimensional object. Because input-output tables are not
symmetric, the eigenvectors arenot orthogonal to each other. In
fact, many eigenvectors are correlated, thereby pickingshocks to
the same groups of sectors. We identify the groups of sectors that
form the fourfactors. The first eigenvector represents shocks to
the heavy manufacturing sectors. Thesecond eigenvector strongly and
negatively correlates with the first and represents threegroups of
industries: (1) most notably, the two sectors relating to agencies,
brokerages, andinsurance; (2) manufacturing of consumer goods; (3)
it has negative entries on the heavymanufacturing industries,
partly neutralizing the shock profile from the first eigenvector.
Thethird eigenvector correlates positively with the second
eigenvector and has positive entrieson the manufacturing of
consumer goods and chemicals. The fourth eigenvector has
close-to-zero correlations with the previous three eigenvectors.
The main sector picked is radioand television broadcasting; in
addition, it also has negative entries on the manufacturingof
chemicals, plastic, and rubber products, partly neutralizing the
shock profiles representedby the third eigenvector. Summarizing, we
find that the welfare impact of any negativetemporary shocks can be
represented by only four (out of 171) eigenvectors.
Our fourth set of results is based on revisiting one of the
earliest historical applicationsof the input-output analysis.
During the World War II, Wassily Leontief was part of thesmall
groups of economist that used input-output analysis for target
selection for strategicbombing (Guglielmo (2008), Harrison (2020)).
We use the input-output table of pre-war NaziGermany and Imperial
Japan to parallel that analysis. First, we provide the list of
sectorsto which temporary shocks generate the largest impact.
Second, we show, for the purposeof finding vulnerability to
temporary shocks, both of these input-output tables also
exhibitlow-dimensional representations: the first three
eigenvectors explain 92% of the variation inwelfare losses for
Japan and 85% for Germany. Third, we demonstrate the over-time
impactof shocks to each sector on every other sector of the
economy, and we show shocks to themetal sectors tend to have
lasting damage across both pre-WWII Germany and Japan.
We now briefly summarize the literature. There is a modern
revival of the literature onproduction networks (see, e.g., reviews
in macroeconomics of Carvalho (2014), Carvalho andTahbaz-Salehi
(2019), and Grassi and Sauvagnat (2019); and, more broadly, Bloch,
Jackson,and Tebaldi (2020) and Jackson, Rogers, and Zenou
(forthcoming)). Acemoglu, Carvalho,Ozdaglar, and Tahbaz-Salehi
(2012) show idiosyncratic sectoral productivity shocks mayhave
aggregate impact. Jones (2011, 2013) develops a model of production
networks withdistortions. A number of recent papers develop various
important aspects of the macroe-conomic implications of the
input-output and production structure of the economy: forexample,
Acemoglu, Akcigit, and Kerr (2015), Oberfield (2018), Baqaee
(2018), Liu (2019),Baqaee and Farhi (2019, 2020), Bigio and La’O
(2020), and Golub, Elliot, and Leduc (2020).
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Unlike these papers, which all feature static models, our main
contribution is to study the dy-namic adjustment of the economy
with the adjustment costs through the lens of the spectralgraph
theory and dynamical system theory.2 We find that temporary shocks
to sectors thatgenerate significant sales through distant linkages
to the consumer are disproportionatelydamaging to the economy. We
also find that shocks to upstream sectors are especially dam-aging
relative to the size of these sectors, precisely because higher
order linkages take a longtime to recover. Our theory also derives
a precise notion of upstreamness that relates to butdiffers from
the upstreamness measure of Antràs, Chor, Fally, and Hillberry
(2012) and thedistortion centrality measure of Liu (2019). Our
analysis of disrupting the Axis economiesis inspired by an
important paper of Davis and Weinstein (2002). The closest in this
aspectto our work is the study of the effects of the 2011 Japanese
earthquake on the supply chainsby Carvalho, Nirei, Saito, and
Tahbaz-Salehi (forthcoming) set in the static framework andan
exceptionally detailed study of the effects of bombing Germany on
resistance to Nazis(Adena, Enikolopov, Petrova, and Voth
(2020)).
2 Model
Our model is a dynamic Cobb-Douglas production network. We
extend the standard, staticproduction network model of Acemoglu et
al. (2012) and Jones (2011, 2013) by introducingdynamic adjustment
costs in input-output linkages. As we show below, allocations in
theequilibrium of the static model coincide with our dynamic
economy’s steady-state but differfrom our transitional path.
There is a representative consumer with exogenous labor supply
¯̀ and N productionsectors that produce from labor and intermediate
inputs. The consumer has utility
V ≡∫ ∞
0
e−ρt ln c (t) dt
where c (t) is a Cobb-Douglas aggregator over sectoral goods j =
1, . . . , N :
c (t) = χc
N∏
j=1
(cj (t))βj ,
N∑
j=1
βj = 1,
where χc ≡∏N
j=1 β−βjj is a normalizing constant. We refer to c (t) as
aggregate consumption
and GDP interchangeably.2In a different setting, Steinerberger
and Tsyvinski (2019) associate a dynamical system with the
static
model of optimal taxation.
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At each time t, the output of production sector i satisfies
qi (t) = χizi (t) (`i (t))αi
N∏
j=1
(mij (t))σij ,
N∑
j=1
σij + αi = 1,
where 0 ≤ αi, σij ≤ 1, χi ≡ α−αii∏N
j=1 σ−σijij is a normalizing constant, zi (t) is sectoral
total factor productivity, li (t) is the amount of labor used,
and mij (t) is the amount of theintermediate good of the sector j
used in the production of the good i.
From now on, wherever it does not cause confusion, we suppress
dependence on time tin the notation.
Our departure from the standard model lies in how intermediate
inputs mij are deliveredfrom seller j to buyer i. In the standard
model, as the economy’s fundamentals change attime t, the prices
and quantities in all connected sectors adjust immediately and
fully totheir new equilibrium levels all at the same time t.
We introduce the concept of time through the transportation of
intermediate inputs acrossproducers. Our formulation captures the
notion that, following temporary negative shocks,the recovery of
input-output linkages must be gradual, and temporary shocks
therefore mayhave lasting impact on the economy. Specifically, to
use input quantity mij at time t, sectori needs to buy
mij × exp (δṁij/mij × 1 (ṁij > 0))
units of input j. The term ṁij ≡ dmij (t) /dt is the rate of
change in the quantity ofintermediate input j used by sector i and
the term 1 (ṁij > 0) states that only the increasesin the goods
use matter. The term exp (δṁij/mij × 1 (ṁij > 0)) captures
sluggish upwardadjustment of inputs and can be interpreted as an
iceberg cost that producer i has to incurwhen it raises the
quantity of input j. The parameter δ captures the ease of
adjustment;when δ → 0, adjustment costs vanish.
The market clearing conditions are
qj = cj +∑
i
mij exp (δṁij/mij × 1 (ṁij > 0)) for all j,
¯̀=∑
i
`i.
For simplicity, we assume goods delivered to the consumer are
not subject to adjustmentcosts, and neither is the use of labor
across production sectors. These choices are madefor expositional
simplicity and are without loss of generality. We can always
accommodateadjustment costs in the purchase of labor or the
consumption good by creating a fictitious
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production sector that buys the consumption bundle and sells to
the consumer or buys laborand sells to other producers.
Equilibrium and Steady State All producers are price-takers.
Let
Ξ (t) ≡ {mij (t) , cj (t) , qj (t) , `j (t) , c (t)}Ni,j=1
be the input-output quantity allocation at time t, let P (t) ≡
{pj (t) , w (t)}Nj=1 be the set ofsectoral prices and of wages at
time t, and let z (t) ≡ {zj (t)}Nj=1 denote the set of time-t
pro-ductivities. Given the initial condition Ξ (0) and the sequence
of sectoral productivity Z (·),an equilibrium is the sequence of
allocation and prices Ξ (t), P (t) such that all producerschoose
input bundles to minimize costs, with
pi =1
ziwαi
(pj exp
(δṁij
/mij × 1 (ṁij > 0)
))σij ,
pjmij exp(δṁij
/mij × 1 (ṁij > 0)
)= σijpiqi,
w`i = αipiqi, pjcj = βjc.
We normalize the consumer price index to one: 1 =∏N
j=1 pβjj . A steady-state equilibrium is
one in which z (t), Ξ (t), P (t) are all time invariant.In what
follows, we use boldface to denote vectors (lower case) and
matrices (upper case).
Let Σ ≡ [σij] denote the matrix of input-output expenditure
shares, and let β denote theN×1 vector of consumption expenditure
shares. Let α be the vector of sectoral value-addedshares. Let γ ′
≡ β′ (I −Σ)−1 be the vector of Domar weights, i.e., sectoral sales
relative toGDP.
Discussion When δ = 0, the economy does not feature adjustment
costs, and the modelbecomes a repeated version of the static
economy in Acemoglu et al. (2012): given the vectorof sectoral TFP
z (t) at each time t, the log-GDP is ln c (t) = const + γ ′ ln z
(t), where γ isthe vector of Domar weights.
When δ > 0, the economy features adjustment costs. However,
allocations and prices inthe steady state of this dynamic model
coincide with those the static equilibrium in Acemogluet al.
(2012). The Hulten’s theorem holds across steady-states: the sales
share γi of sectori characterizes the steady-state importance of
each sector’s TFP. Specifically, let css denotethe steady-state
consumption per period; then
ln css = const + γ ′ ln z.
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In our formulation, input usage is slow to expand but may shrink
instantaneously. Thisis for expositional simplicity; the model can
be easily generalized to accommodated sluggishdownward adjustment
of inputs as well.
3 Slow recovery from a temporary TFP shock
Consider a production network affected by temporary negative TFP
shocks to some sectors.These shocks reduce sectoral production and
may propagate through input-output linkagesand affect output in
other sectors. After these negative shocks revert, how quickly does
theeconomy recover? We show, when production linkages take time to
recover, the topology ofa production network is a key determinant
of its resilience to negative shocks.
Figure 1: Two stylized example networks
1
2
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(a) A horizontal production economy (b) A vertical production
chain
To illustrate the intuition, consider two example networks. In
Figure 1, panel (a) showsthe network structure of a horizontal
production economy. Here, labor is the only factorof production in
each sector i ∈ {1, . . . , N}, and sectors i does not use any of
the goodsj ∈ {1, . . . , N} in production. All of the goods are
part of the consumption bundle c. Sincethere are no input-output
linkages, the adjustment costs in this setting are irrelevant, anda
negative TFP shock to zi has zero impact on the economy once the
TFP reverts back,measured by either sectoral output or GDP.
Now consider panel (b), which shows the network structure of a
vertical production chain.Here, labor is the only factor of
production of sector 1, and each subsequent sector i usesinputs
only of the sector i − 1. Only good N is used in final consumption
c. Consider atemporary decline in sector 1’s productivity z1.
Sector 1’s output declines for the duration ofthe negative shock;
moreover, because sector 2 requires good 1 as inputs, the output of
sector2 declines as well, and in fact output declines in all
sectors i ∈ {1, . . . , N}. After the initialTFP shock disappears
and as z1 reverts, output in sector 1 recovers immediately.
However,because of adjustment costs in the recovery of input-output
linkages, sectoral output forall i ≥ 2 may stay extendedly
depressed, and the economy as a whole—measured by the
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consumption aggregator c (t), i.e., the GDP—may take a long time
to recover. By contrast,a temporary reduction in sector N ’s TFP zN
has no lasting impact on the economy, whichrecovers immediately
after the shock dissipates. More generally, in the vertical
networkof panel (b), the economy recovers more slowly from negative
shocks that affect relativelyupstream sectors.
We now formalize the analysis, and we analyze sectoral
susceptibility in a productionnetwork from our dynamic
perspective.
3.1 Negative shocks and transitional dynamics
We analyze an economy initially in a steady-state with sectoral
log-productivities {ln zi}Ni=1,and we consider temporary, negative
TFP shocks that reduce sectoral productivities attime zero to {ln
zi − z̃i}Ni=1. We use z̃i > 0 to denote the absolute value in
logs of thenegative shocks for sector i. For expositional
simplicity, we assume sectoral TFP revertsback instantaneously to
the pre-shock steady-state levels {zi}Ni=1. We use t = 0− and t =
0to respectively index the time at and after the negative TFP
shocks. When the negativeshocks are present at t = 0−, log-GDP
declines by γ ′z̃ relative to its steady-state level,where γ ′ is
the Domar weights, consistent with Hulten (1978) and Acemoglu et
al. (2012).We now analyze the dynamic path of sectoral output and
GDP during the recovery, fromt = 0 onwards.
Even as sectoral TFP recovers at t = 0, the use of intermediate
inputs can only growgradually over time and cannot jump
instantaneously. Hence, sectoral output increasesexactly in
proportion to the TFP recovery, and the total output in sector j
exceeds thetotal quantity of good j used as production inputs. The
excess output is dispensed as theadjustment costs required to
expand input j for the future. With passage of time,
sectorscontinue to expand the use of inputs, sectoral output
continues to expand even though TFPis constant. Eventually the
economy converges back to the initial steady-state as t→∞.
To solve for the transition path, let
xj (t) ≡ lnqj (t)− cj (t)∑
imij (t)
be the log-ratio between the total quantity of good j supplied
to and used by other producers.The ratio qj(t)−cj(t)∑
imij(t)is equal to one in a steady-state, and xj = 0 for all j.
Away from a steady-
state, the ratio captures the proportional adjustment costs
incurred for expanding input jin production. Because all producers
i spend a constant share of input expenditure on goodj along the
transition path, xj (t) = δṁij
/mij for all i, and δ−1xj (t) captures the rate at
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which all sectors expand their use of input j.
Proposition 1. Laws of Motion. Consider a TFP shock vector z̃
that affects the steady-state economy at time zero and reverts back
instantaneously. The law of motion for sectoraloutput vector q
is
d ln q
dt= δ−1Σx (t) , with the initial condition ln q (0) = ln qss −Σ
(I −Σ)−1 z̃, (1)
where I is the identity matrix and Σ ≡ [σij] is the matrix of
input-output coefficients.The law of motion for GDP is
d ln c
dt= δ−1β′Σx (t) , with the initial condition ln c (0) = ln css −
β′Σ (I −Σ)−1 z̃.
The dynamic path of x satisfies the ODE system
dx (t)
dt= − (I −Σ)x (t) , with the initial condition x (0) = z̃.
To understand this Proposition, first suppose the negative TFP
shocks were permanent.Output declines in sectors directly affected
by the shocks. Moreover, because of productionlinkages, output also
declines in sectors that purchase inputs—directly or
indirectly—fromthe shocked sectors. The total impact of negative
shocks on sectoral output is captured by− (I −Σ)−1 z̃, where the
Leontief inverse (I −Σ)−1 ≡ I+Σ+Σ2+ · · · captures the
infiniterounds of higher order effects through input-output
linkages. This is indeed the finding ofAcemoglu et al. (2012) and
Acemoglu et al. (2015) in the standard, static production
networkmodel.
This Proposition 1 instead pertains to temporary shocks. As
sectoral TFP recoversinstantaneously, log-output directly recovers
by z̃; hence, at time t = 0, sectoral outputsatisfies
ln q (0) = ln qss︸ ︷︷ ︸initial steady state
− (I −Σ)−1 z̃︸ ︷︷ ︸effect of permanentnegative shocks
+ z̃︸︷︷︸recovery of TFP
.
The input-output linkages destroyed by the negative shocks take
time to recover. Becauseδ−1xj (t) = ṁij/mij captures the rate at
which all producers expand their use of input j, theoutput of
sector i grows at rate
q̇i/qi = δ−1
N∑
j=1
σijxj (t) ,
10
-
and in the vector formd ln q
dt= δ−1Σx (t) .
The law of motion for GDP follows from the fact that ln c (t)−
ln css = β′ (ln q (t)− ln qss)for all t.
Finally, to derive the law of motion for x (t), note
dxj (t)
dt=
d ln qjdt
− d ln (∑
imij (t))
dt
captures the difference between the rate at which sectoral
output expands (the first term)and the rate at which sectoral goods
are used as intermediate inputs (the second term); thelatter is
equal to δ−1xj (t). In the vector form,
dx
dt= δ−1Σx− δ−1x.
The ODE system for x (t) has an explicit solution in terms of
the matrix exponential:
x (t) = e−δ−1(I−Σ)tz̃, (2)
where matrix exponential for any generic matrix M is defined
as
eM ≡∞∑
k=0
M k
k!.
Intuitively, immediately after the negative TFP shock recedes, x
(0) = z̃. As productionlinkages recover over time and as the
economy converges back to the steady-state, x (t)converges to the
zero vector. The term δ modulates the rate of convergence; the
systemconverges at a faster rate if adjustment cost δ is small. The
next proposition describes thetime paths or the flow of the
sectoral outputs and consumption.
Proposition 2. Flow of Output and Consumption. The flow of
sectoral output satisfies
ln q (t) = ln qss −Σ (I −Σ)−1 e−δ−1(I−Σ)tz̃
and the flow of aggregate consumption satisfies
ln c (t) = ln css − β′Σ (I −Σ)−1 e−δ−1(I−Σ)tz̃.
Proof. We have
11
-
ln q (t) = q (0) + δ−1Σ
∫ t
0
x (s) ds
= q (0) + δ−1Σ
[∫ t
0
e−δ−1(I−Σ)sds
]z̃
= ln qss − (I −Σ)−1 z̃ + z̃︸ ︷︷ ︸q(0)
+ Σ (I −Σ)−1(I − e−δ−1(I−Σ)t
)z̃
= ln qss −Σ (I −Σ)−1 e−δ−1(I−Σ)tz̃.
The expression for c (t) is derived analogously.
When productivity in sector j recovers, the sector’s output
expands immediately, whichgradually translates into the expansion
of input j used in other sectors i, thereby causing i’soutput to
expand gradually over time. The vector
(− (I −Σ)−1 z̃ + z̃
)= −Σ (I −Σ)−1 z̃
captures the extent to which log-sectoral outputs at t = 0 are
below their steady-state levels;it can be re-written as
−Σ (I −Σ)−1 z̃ = −( ∞∑
s=1
Σs
)z̃
where each successive term in the summation captures a higher
round of input-output link-ages to be recovered from the initial
shock. The expression
(−Σ (I −Σ)−1 e−δ−1(I−Σ)t
)is
the log-deviation in output relative to steady-state levels at
time t; it is the continuous timeanalogue of the discrete partial
sum −∑∞s=t Σs. By varying t, the expression captures thefact that
general equilibrium linkages recover gradually, and higher rounds
of linkages takelonger to recover.
The rate of recovery is inversely related to δ. As δ → 0, the
convergence towards thesteady-state becomes instantaneous:
limδ→0
(−Σ (I −Σ)−1 e−δ−1(I−Σ)t
)= 0 for any t > 0.
More broadly, this proposition shows that the properties of the
dynamical system de-scribed by the gradual adjustment of the
economy are tightly related to the properties ofthe input-output
matrix via the sequence of its powers Σs. The parameter δ modulates
thespeed of adjustment and thus makes the dynamical system that we
study somewhat broaderthan the classical dynamical systems that are
associated with a given matrix A. There, atypical dynamical system
is given by ḃ = Ab (Colonius and Kliemann (2014)).
12
-
3.2 Sectoral shocks and welfare
We now characterize the impact of sectoral TFP shocks on
consumer welfare. Let V ss denoteconsumer welfare in the initial
steady state absent the TFP shock.
Proposition 3. Welfare Impact of Temporary TFP Shocks. Let
v′ ≡ 1ρ
[β′ (I −Σ)−1 − β′
(I − Σ
1 + ρδ
)−1].
The impact of temporary, negative TFP shocks z̃ on welfare
is
V (z̃)− V ss =∫ ∞
0
e−ρs (ln c (s)− ln css) ds = −v′z̃.
Proof. We have
V (z̃)− V ss0 =∫ ∞
0
e−ρs (ln c (s)− ln css0 ) ds
= −β′Σ (I −Σ)−1∫ ∞
0
e−δ−1((1+ρδ)I−Σ)tdtz̃
= −δβ′Σ (I −Σ)−1 ((1 + ρδ) I −Σ)−1 z̃= −1
ρβ′Σ
[(I −Σ)−1 − ((1 + ρδ) I −Σ)−1
]z̃
= −1ρ
[β′ (I −Σ)−1 − β′
(I − Σ
1 + ρδ
)−1]z̃.
The vector v′ captures the elasticity of welfare to temporary,
negative TFP shocks. Whenδ = 0, recovery is instantaneous, and
temporary shocks have no impact on consumer welfare.The first term,
1
ρβ′ (I −Σ)−1, is proportional to the sectoral Domar weight and
captures the
impact on welfare of permanent negative TFP shocks. The second
term 1ρβ′(I − Σ
1+ρδ
)−1
captures the impact of slow recovery of input-output linkages.
One can also think about thisterm as solving for χ′ in the
following expression:
β′ +1
1 + ρδΣχ′ = χ′,
which is identical to the expression for the sectoral Domar
weights (γ ′ solves β′+ Σγ ′ = γ ′)adjusted by the factor 1
1+ρδ.
13
-
It is informative to rewrite v′ as
v′ =1
ρβ′
∞∑
s=0
(1− (1 + ρδ)−s
)Σs (3)
and compare the expression with sectoral Domar weights:
γ ′ = β′∞∑
s=0
Σs. (4)
In a static model, the Domar weight captures the impact of
sectoral TFP on aggregateconsumption, and each term β′Σs in the
power series captures the s-th round of networkeffect: β′ captures
the first round, direct effect of TFP on consumption, β′Σ captures
theindirect effect of sectoral TFP on other producers who supply to
the consumer, and so on.The Domar weight is also equal to a
sector’s sale relative to GDP, and each term β′Σs inthe power
series captures the revenue from the s-th round indirect sales to
the consumer.
In our dynamic model, temporary shocks may have lasting effect
on output and welfareprecisely because of higher-order linkages Σs,
s > 0. When δ > 0, input-output linkages areslow to recover,
and
(1− (1 + ρδ)−s
)Σs captures the present discounted value of consump-
tion affected by the slow recovery of the s-th order linkages.
Effectively, the power seriesin (3) disproportionately removes the
initial entries in (4) while keeping the tail entriesunchanged:
v′ =1
ρβ′[(1− 1) Σ0 +
(1− (1 + ρδ)−1
)Σ1 +
(1− (1 + ρδ)−2
)Σ2 + ...
].
Note the weight on Σ0 is 0 and the weight on Σs converges to 1
as s→∞.We now summarize this proposition as our first main result
of the paper: in the presence
of adjustment frictions, shocks to sectors that generate
significant sales through distantlinkages to the consumer are
disproportionately damaging to the economy. These shockshave large
and lasting impact on GDP even as sectoral TFP recovers.
Alpha centrality and global versus local influence We next show
that the welfareimpact measure v′ also can be connected to a
measure of centrality, alpha centrality, ina network represented by
the input-output matrix. The alpha centrality for α ∈ (0, 1]
isdefined as:
ι′α ≡ β′(I − αΣ)−1.
14
-
Intuitively, this is a centrality measure where a parameter α is
used to weigh the higherorder input-output linkages, represented by
the powers of the matrix Σ:
ι′α ≡ β′[Σ0 + αΣ1 + α2Σ2 + ...
].
The i-th entry in β′Σs captures the sales (relative to GDP) that
sector i generates throughs rounds of linkages before reaching the
final consumer.
A related way to think about centrality is in terms of a random
walk on the network,where Σij is the probability of reaching j from
i in one walk. The ij-th entry in Σs thenmeasures the probability
of reaching j from i in the walks of length s. As parameter (α ≤
1)decreases, shorter walks become more important, and local
influences carry higher signifi-cance. When α increases, longer
walks become more important, and global influences carryhigher
significance. In the limit case as α → 1, the walks of any length
carry identicalweights, and the alpha centrality measure becomes
the Domar weight. In this sense, alphacentrality tunes between
rankings based on short walks (local influence) and those based
onlong walks (global influence) (Benzi and Klymko (2015)).
The welfare v′ is thus proportional to the difference in the
alpha centralities ι′α1 − ι′α1 ,where α1 = 1 and α2 = (1 + ρδ)
−1 . Now, let us slightly modify the notion of alpha
centralityby defining it as
ι̃′ ≡ β′[a0Σ
0 + a1Σ1 + a2Σ
2 + ...],
for some sequence {a0, a1 . . . }. Assuming that such weighted
power series converge, thismeasure weights the walks of length k
with the parameter ak. In the case of alpha centralitywith α <
1, ak = αk and is geometrically decreasing from a0 = 1 and a∞ = 0.
Thewelfare measure v′ is also a (modified version of) alpha
centrality with ak = 1−αk2 and thusincreasing between a0 = 0 and a∞
= 1. One can thus think of it as being conceptually similarto the
usual alpha centrality, where the welfare measure, however,
relatively prioritizes thelonger walks or higher order input output
linkages and thus the global over local influences.
The term (1 + ρδ)−1 also defines a one-parameter family of the
economies that can bethought of as a multi-scale representation of
the static input output matrix. Specifically, thespeed of
adjustment and the discount factor of the agent determine the
scale—the relativeimportance of the higher-order links and thus the
importance of the global versus localstructures.
Vertical Economy Revisited. It is now instructive to revisit the
examples in Figure 1.In the horizontal economy of panel (a), there
are no input-output linkages; consequently, v isthe zero vector,
and temporary shocks that recover instantaneously have zero impact
on this
15
-
economy. By contrast, temporary shocks may have lasting impact
in the vertical economyof panel (b), with the network diagram
reproduced below, along with input-output table ofthis economy.
Sector 1 is the most upstream and sector N is the most
downstream.1
2
N
...
AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRS0WPRi8cKthbaUDabTbt2sxt2J4VS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBXcoOd9O4W19Y3NreJ2aWd3b/+gfHjUMirTlDWpEkq3Q2KY4JI1kaNg7VQzkoSCPYbD25n/OGLacCUfcJyyICF9yWNOCVqp1R1FCk2vXPGq3hzuKvFzUoEcjV75qxspmiVMIhXEmI7vpRhMiEZOBZuWuplhKaFD0mcdSyVJmAkm82un7plVIjdW2pZEd67+npiQxJhxEtrOhODALHsz8T+vk2F8HUy4TDNkki4WxZlwUbmz192Ia0ZRjC0hVHN7q0sHRBOKNqCSDcFffnmVtC6qfq16eV+r1G/yOIpwAqdwDj5cQR3uoAFNoPAEz/AKb45yXpx352PRWnDymWP4A+fzB84Hj0o=
C
L
12
N. . .
AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRS0WPRi8cKthbaUDabTbt2sxt2J4VS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBXcoOd9O4W19Y3NreJ2aWd3b/+gfHjUMirTlDWpEkq3Q2KY4JI1kaNg7VQzkoSCPYbD25n/OGLacCUfcJyyICF9yWNOCVqp1R1FCk2vXPGq3hzuKvFzUoEcjV75qxspmiVMIhXEmI7vpRhMiEZOBZuWuplhKaFD0mcdSyVJmAkm82un7plVIjdW2pZEd67+npiQxJhxEtrOhODALHsz8T+vk2F8HUy4TDNkki4WxZlwUbmz192Ia0ZRjC0hVHN7q0sHRBOKNqCSDcFffnmVtC6qfq16eV+r1G/yOIpwAqdwDj5cQR3uoAFNoPAEz/AKb45yXpx352PRWnDymWP4A+fzB84Hj0o=
C
L
along with input-output table of this economy. Sector 1 is the
most upstream and sector N is themost downstream.1
2
N
...
AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRS0WPRi8cKthbaUDabTbt2sxt2J4VS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBXcoOd9O4W19Y3NreJ2aWd3b/+gfHjUMirTlDWpEkq3Q2KY4JI1kaNg7VQzkoSCPYbD25n/OGLacCUfcJyyICF9yWNOCVqp1R1FCk2vXPGq3hzuKvFzUoEcjV75qxspmiVMIhXEmI7vpRhMiEZOBZuWuplhKaFD0mcdSyVJmAkm82un7plVIjdW2pZEd67+npiQxJhxEtrOhODALHsz8T+vk2F8HUy4TDNkki4WxZlwUbmz192Ia0ZRjC0hVHN7q0sHRBOKNqCSDcFffnmVtC6qfq16eV+r1G/yOIpwAqdwDj5cQR3uoAFNoPAEz/AKb45yXpx352PRWnDymWP4A+fzB84Hj0o=
C
L
12
N
. . .
AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRS0WPRi8cKthbaUDabTbt2sxt2J4VS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MBXcoOd9O4W19Y3NreJ2aWd3b/+gfHjUMirTlDWpEkq3Q2KY4JI1kaNg7VQzkoSCPYbD25n/OGLacCUfcJyyICF9yWNOCVqp1R1FCk2vXPGq3hzuKvFzUoEcjV75qxspmiVMIhXEmI7vpRhMiEZOBZuWuplhKaFD0mcdSyVJmAkm82un7plVIjdW2pZEd67+npiQxJhxEtrOhODALHsz8T+vk2F8HUy4TDNkki4WxZlwUbmz192Ia0ZRjC0hVHN7q0sHRBOKNqCSDcFffnmVtC6qfq16eV+r1G/yOIpwAqdwDj5cQR3uoAFNoPAEz/AKb45yXpx352PRWnDymWP4A+fzB84Hj0o=
C
L
Σ =
0 0 · · · 0 01 0 · · · 0 00 1 · · · 0 0...
. . ....
0 0 · · · 1 0
, β =
0
0...0
1
.
By construction, the Domar weight is identically one for all
sectors, γ ′ ≡ β′ (I −Σ)−1 = 1′.TFP shocks in every sector has
identical impact on GDP in a static model. In our dynamic
economy,however, the welfare impact of sectoral shocks is no longer
constant; in fact, the impact of temporaryshocks follow
v′ = ρδ[
1−∑Ns=1 (1 + ρδ)−s , 1−∑N−1
s=1 (1 + ρδ)−s , · · · , 1−
(1
1+ρδ +(
11+ρδ
)2), 1− 11+ρδ , 0
]
Hence, temporary shocks to sector i are more damaging than to
sector j > i, despite all sectorshaving the same Domar
weight.
Connection to Measures of Upstreamness An economy is more
susceptible to temporaryshocks that hit sectors with signifcant
sales through distant linkages to the consumer. Undera general
network structure—including the real-world input-output tables we
investigate in latersections—the welfare elasticity to sectoral
shocks tends to be higher if the sector is large and ismore
upstream.
In fact, the welfare elasticity vi can be written as the product
between sector i’s Domar weightand a notion of upstreamness. Recall
from (4) that the sectoral Domar weight can be written asγ ′ =
β′
∑∞s=0 Σ
s, where the i-th component of β′Σs captures the sales of sector
i (relative to GDP)that reaches the final consumer through s-rounds
of input-output linkages. Antras et al. (2012)defines an
“upstreamness” measure Upi that satisfies:
Upi = 1 ·βiγi
+ 2 · [β′Σ]iγi
+ 3 ·[β′Σ2
]i
γi+ 4 ·
[β′Σ3
]i
γi+ · · ·
=
∞∑
s=0
as · [β′Σs]iγi
, with as = s+ 1.
9
In this vertical economy, each successive power of the
input-output matrix contains asmaller identity sub-matrix in the
bottom-left and zeros otherwise, and the Leontief-inverseis a
lower-triangular matrix of ones:
Σs =
[0s×(N−s) 0s×s
I(N−s)×(N−s) 0(N−s)×s
], (I −Σ)−1 =
1 0 · · · 0 01 1 · · · 0 01 1
. . . 0 0... 1 . . . 1
...1 1 · · · 1 1
.
For example, when N = 4,
In this vertical economy, each successive power of the
input-outpout matrix contains asmaller identity submatrix in the
bottom-left and zeros otherwise, and the Leontief-inverseis a
lower-triangular matrix of ones:
Σs−1 =
[0(N−s)×s 0s×(N−s)
Is×s 0(N−s)×(N−s)
], (I −Σ)−1 =
1 0 · · · 0 01 1 · · · 0 01 1
. . . 0 0... 1 . . . 1
...1 1 · · · 1 1
.
[I think we can delete the math expressions above if we keep the
example] For example, whenN = 4,
Σ =
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
, Σ2 =
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
, Σ3 =
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
, (I −Σ)−1 =
0 0 0 0
1 0 0 0
1 1 0 0
1 1 1 0
.
By construction, the Domar weight is identically one for all
sectors, γ ′ ≡ β′ (I −Σ)−1 = 1′.TFP shocks in every sector has
identical impact on GDP in a static model. In our dynamiceconomy,
however, the welfare impact of sectoral shocks is no longer
constant; in fact, theimpact of temporary shocks follow
v′ = ρδ
[1−∑Ns=1 (1 + ρδ)
−s , · · · , 1−(
11+ρδ
+(
11+ρδ
)2), 1− 1
1+ρδ, 0
]
Hence, temporary shocks to sector i are more damaging than to
sector j > i, despite allsectors having the same Domar
weight.
Connection to Katz Centrality and Upstreamness Temporary shocks
are damagingif they hit sectors with significant sales through
distant linkages to the consumer. We nowshow vi is the product
between sector i’s Domar weight and its Katz (1953) centrality of
theinput-output supply network. In our context, Katz centrality is
a measure of upstreamness:it captures the network-adjusted distance
of sectoral supply to the final consumer. Hence,temporary shocks
are more damaging to the economy if they hit large sectors that are
alsoupstream and supply disproportionate fractions of outputs to
other upstream producers.
Let ηi ≡ vi/γi be the welfare impact of a temporary shock to
sector i relative to the
sectoral size. Let Θ be the input-output supply matrix, whose
in-th entry is σniγn/γi, i.e.the fraction of revenue that sector i
derives from selling to sector n. Intuitively, the entries
15
By construction, the Domar weight is identically one for all
sectors, γ ′ ≡ β′ (I −Σ)−1 =1′. TFP shocks in every sector has
identical impact on GDP in a static model. In ourdynamic economy,
however, the welfare impact of sectoral shocks is no longer
constant; infact, the impact of temporary shocks follow
v′ = ρδ[
1− (1 + ρδ)−N , · · · , 1−(
11+ρδ
)2, 1− 1
1+ρδ, 0
]
Hence, temporary shocks to sector i are more damaging than to
sector j > i, despite allsectors having the same Domar
weight.
Figure 2 shows the path of GDP over time when each sector in the
vertical economy (withN = 4) is affected by a temporary, negative
TFP shock. As the figure demonstrates, shocksto relatively upstream
sectors have long-lasting effects: the economy takes the longest
time
16
-
Figure 2: Time path of GDP losses from sectoral shocks in the
vertical economy
0
time
0
outp
ut lo
ss
sector 4
sector 3
sector 2
sector 1
to recover from shocks to sector 1—the most upstream—and
recovers instantaneously fromshocks to sector 4. Consequently, v1
> v2 > v3 > v4 as the measure v′ integrates the entirepath
of output losses, discounting the future at rate ρ.
Connection to Katz Centrality and Upstreamness Temporary shocks
are damagingif they affect the sectors with significant sales
through distant linkages to the consumer. Wenow show vi is the
product between sector i’s Domar weight and its Katz (1953)
centralityof the input-output revenue share matrix. In our context,
Katz centrality is a measureof upstreamness: it captures the
network-adjusted distance of sectoral supply to the finalconsumer.
Hence, temporary shocks are more damaging to the economy if they
affect largesectors that are also upstream and supply
disproportionate fractions of outputs to otherupstream
producers.
Let ηi ≡ vi/γi be the welfare impact of a temporary shock to
sector i relative to
the sectoral size. Let Θ be the input-output revenue share
matrix, whose in-th entry isθin ≡ σniγn/γi, i.e. the fraction of
revenue that sector i derives from selling to sector n.
In-tuitively, the entries of the expenditure share matrix Σ are
obtained by dividing the value ofintermediate inputs by the sales
of the buyer, whereas entries of Θ are obtained by dividingthe
value of inputs by the sales of the supplier.
Proposition 4. Welfare impact is the Domar weight times the Katz
centrality.
η = δ
[ ∞∑
s=1
(1
1 + ρδΘ
)s]1.
17
-
Proof. Let r ≡ 11+ρδ
and x′ ≡ β′ (I − rΣ)−1. We have
ρηj = 1− xj/γj = 1− βj/γj − r∑
i
σijxi/γj =∑
i
θji − r∑
i
θji (1− ρηi)
=⇒ η = δrΘ1 + rΘη = δ (I − rΘ)−1 rΘ1 = δ[ ∞∑
s=1
(1
1 + ρδΘ
)s]1.
Katz centrality can be re-written implicitly as η = δ1+ρδ
Θ1 + 11+ρδ
Θη, or, in scalar form,
ηi =δ
1 + ρδ
N∑
n=1
Θin +1
1 + ρδ
N∑
n=1
Θinηn.
The first term on the right-hand-side is a constant ( δ1+ρδ
) times the total fraction of sectori’s output supplied to
intermediate producers (rather than the consumer). The second
termis the average Katz centrality of the producers that use good i
as inputs, weighted by thefraction of i’s output sold to each
buyer, and scaled down by a factor 1
1+ρδ. Hence, a sector
is Katz-central if it supplies a disproportionate fraction of
output to other Katz-centralproducers.
Katz centrality is a natural notion of upstreamness.3 Recall
from (4) that the sectoralDomar weight can be written as γ ′ =
β′
∑∞s=0 Σ
s, where the i-th component of β′Σs capturesthe sales of sector
i (relative to GDP) that reaches the final consumer through
s-rounds ofinput-output linkages. Antràs et al. (2012) defines an
upstreamness measure that capturesthe average number of rounds it
takes for sectoral output to reach the final consumer:
Upi = 1 ·βiγi
+ 2 · [β′Σ]iγi
+ 3 · [β′Σ2]iγi
+ · · · =∞∑
s=0
as · [β′Σs]iγi
, with as = s+ 1.
More generally,∑∞
s=0as·[β′Σs]i
γiis a measure of sector i’s upstreamness for any increasing
and
convergent sequence {as}∞s=0 because such a sequence up-weights
sectoral sales that are moredistant to the consumer. Katz
centrality can also be written in this form using the sequenceas =
ρ
−1 (1− (1 + ρδ)−s).
3The Katz centrality is also isomorphic to the distortion
centrality of Liu (2019) in a production networkwith constant
market imperfection wedges.
18
-
4 Eigendecomposition of the Input-Output Matrix and
the Dynamical System
Temporary shocks to large and upstream sectors are
disproportionately damaging becauseinput-output linkages are slow
to recover from these shocks. We now examine determinantsof the
welfare measure v′ from the spectral point of view.
Consider the diagonalization of the input-output table, Σ = UΛW
, where Λ is a diagonalmatrix of eigenvalues {λk}Nk=1, andW = U−1.
The columns of U are the right-eigenvectors,and the rows ofW are
the corresponding left-eigenvectors. Assuming Σ is full-rank, U
andW both span the N -dimensional complex coordinate space CN and
are therefore are bothbases of the space. Without loss of
generality, we arrange the eigenvalues and eigenvectorsin
decreasing absolute value, with |λ1| ≥ |λ2| ≥ · · · ≥ |λN |. Note
that since Σ is row-sub-stochastic (i.e., the sum of intermediate
expenditure shares must be ≤ 1 in all sectors, withstrict
inequality for at least some sectors), the dominant eigenvalue must
have absolute valuebelow one, i.e., |λ1| < 1.
Right Eigenvectors. Let uk denote the k-th right-eigenvector,
i.e., the k-th column ofthe matrix U . This is the vector that,
when being multiplied by Σ on the left, becomes ascaled version of
itself:
Σsuk = λskuk for all s ∈ Z≥0.
Now consider a negative TFP shock vector that equals to (−uk),
and suppose uk ∈ RN . Thefirst round effect lowers sectoral output
by (−uk); the second round effect lowers sectoraloutput by (−Σuk =
−λkuk); the third round effect lowers sectoral output by (−λ2kuk),
andso on. That is, at each round of propagation, the productivity
shock vector (−uk) alwaysreduces sectoral output in proportion to
−uk, with effects scaled-down by a factor equal tothe eigenvalue λk
relative to the previous round. In other words, uk is the profile
of TFPshocks with every round of general equilibrium effect always
in proportion to the first roundbut decays at rate λk after each
round of propagation. We construct the right-eigenvectorsso that
2-norm of uk is equal to 1 for all k.
Complex Eigenvalues. In general, the eigenvalues and
eigenvectors can be complex-valued. Complex eigenvalues exist in
conjugate pairs: when an eigenvalue λk is complex—inwhich case uk
must be complex too—the conjugate transpose λk is also an
eigenvalue, andthe corresponding eigenvector is uk. Now consider a
shock profile z̃ ≡ Re (uk). Note
19
-
Re (uk) = uk+uk2 , and following equation must be satisfied:
ΣsRe (uk) = Re (λskuk) ,
where the operator Re (·) selects the real part of a complex
vector. Note that, when λk iscomplex, Re (λskuk) 6= Re (λk) ·Re
(λs−1k uk
); hence, the shock vector Re (uk) is not a scaled-
down version of itself when left-multiplied by the input-output
table Σ. The higher roundsof network effects from the shock vector
Re (uk) no longer decay to zero at a constant rateRe (λk); instead,
complex eigenvalues introduce oscillatory motion in the impact of
negativeshocks as the network effects converge to zero under higher
and higher rounds. In otherwords, if we project the network effects
Σsz̃ onto an N -dimensional vector space, the higherrounds of
network effects (higher s) associated with a real eigenvector shock
profile convergeto zero following a straight line connecting uk and
the origin. In contrast, the networkeffects of a shock profile Re
(uk) with a complex eigenvalue would converge to zero followingan
elliptical spiral.
For expositional purposes, we focus on real-valued eigenvalue
and eigenvectors. As weshow below, the largest (hence, as we show
below, more important) eigenvalues of the real-world input-output
tables are all real. Moreover, the imaginary components of any
complexeigenvalues are overall significantly smaller than the real
components, implying that oscilla-tory higher-order network effects
are small relative to the effects that decays exponentially.
Left Eigenvectors. Letw′k denote the k-th left-eigenvector,
i.e., the k-th row of the matrixW . W is the matrix that projects
sectoral shocks onto the right-eigenspace. Specifically, anyTFP
shock vector z̃ can be written as a linear combination {ak}Nk=1 of
the right-eigenvectors
z̃ =N∑
k=1
akuk,
or, in matrix notation,z̃ = Ua′.
The vector a′ can be obtained by
Wz̃ = WUa′ = U−1Ua′ = a′.
That is, ak = w′kz̃, or in vector form, a′ = Wz̃.To summarize,
the right-eigenvectors capture the shock profiles whose general
equilib-
rium impact decays at rate governed by the corresponding
eigenvalues; the left-eigenvectors
20
-
convert shock profiles into coordinates in the right-eigenspace.
We sometimes refer to theright-eigenvectors simply as
eigenvectors.
Eigen-Decomposition of Domar Weights andWelfare Impact We now
use the basesU and W to further decompose the aggregate impact of
sectoral shocks.
Proposition 5. Eigen-Decomposition of Domar Weights and Welfare
Impact. TheDomar weight can be written as
γ ′ = β′N∑
k=1
1
1− λkukw
′k. (5)
The vector v′ can be written as
v′ = δβ′N∑
k=1
λk(1− λk) (1 + ρδ − λk)
ukw′k. (6)
Proof. Consider the Domar weight
γ ′ = β′( ∞∑
s=0
Σs
)= β′U
( ∞∑
s=0
Λs
)W
= β′N∑
k=1
( ∞∑
s=0
λsk
)ukw
′k = β
′N∑
k=1
1
1− λkukw
′k.
The welfare impact
v′ =1
ρβ′
∞∑
s=0
(1− (1 + ρδ)−s
)Σs
=1
ρβ′
N∑
k=1
(1
1− λk− 1
1− 11+ρδ
λk
)ukw
′k
= δβ′N∑
k=1
λk(1− λk) (1 + ρδ − λk)
ukw′k.
The proposition turns the infinite-sum-of-power-series
representation of γ ′ and v′ in (4)and (3) into finite sums over
eigen components.
21
-
To understand the implication of Proposition 5, first consider a
TFP shock profile cap-tured by z̃ = uk. Note that
w′`uk =
1 if ` = k
0 otherwise.
The shock profile uk’s impact in a static model is therefore
captured by
γ ′uk = β′N∑
`=1
1
1− λ`u`w
′`uk
=1
1− λkβ′uk.
That is, the shock uk affects static consumption only through
the k-th eigen component,with the direct effect being β′uk, the
s-th round indirect network effect being λskβ′uk, anda cumulative
effect of
∑∞s=0 λ
skβ′uk =
11−λkβ
′uk.We now analyze (v′uk), i.e., the welfare impact of the
temporary shock vector uk in our
dynamic economy. Since the shock vector uk affects consumption
at all times only throughthe k-th eigen component, the impact can
be re-written as
v′uk =1
ρβ′uk
( ∞∑
s=0
(1− (1 + ρδ)−s
)λsk
).
The additional term(1− (1 + ρδ)−s
)assigns zero weight to the direct effect of the shock (s =
0)—because TFP recovers at t = 0—and an increasing sequence of
weights to higher-ordernetwork effects. The cumulative effect is
then scaled by 1/ρ to reflect the fact that we have adynamic
economy with consumer discount rate ρ. The expression ρ−1
(∑∞s=0
(1− (1 + ρδ)−s
)λsk)
further simplifies to δ λk(1−λk)(1+ρδ−λk) as the proposition
shows.
Any generic TFP shock vector z̃ can be projected onto the
right-eigenspace with z̃ =∑Nk=1 ukak, and ak ≡ w′kz̃ is its k-th
coordinate after the projection. The overall effect on
welfare is −v′z̃ = −v′∑Nk=1 ukak.As we show below,
quantitatively v′ has a low-dimensional factor representation,
where
v′z̃ can be approximated closely by its projection onto the
first K (K = 4) eigenvectors:
v′z̃ ≈ v′K∑
k=1
ukak.
Proposition 5 shows why this is the case and also why the Domar
weight does not have agood approximation in a low-dimensional
sub-eigenspace. To see this, consider two distinct
22
-
shock profiles captured by real right-eigenvectors uk and u`
with |λk| < |λ`|. Let � ≡ |β′uk||β′u`|
denote the relative loadings of the consumption share vector on
these two eigenvectors. Therelative impact between the eigenvector
shock profiles on aggregate consumption in the staticmodel is
|γ ′uk||γ ′u`|
=1− λ`1− λk
× �
On the other hand, their relative impact in our dynamic model
is
|v′uk||v′u`|
=|λk| (1 + ρδ − λ`)|λ`| (1 + ρδ − λk)︸ ︷︷ ︸
-
costs significantly down-weight the direct and initial rounds of
network effects, v′ may havea factor representation as long as |λk|
declines relatively fast in k.
5 Factor Structure of the U.S. Input-Output Table
We now turn to the 2012 U.S. input-output table published by the
U.S. Bureau of La-bor Statistics. We show that the high-dimensional
input-output table—171 by 171 sectorsunder broad categories of
agriculture, mining, manufacturing, and services4—has a
low-dimensional, 4-factor structure in terms of its susceptibility
to temporary shocks: the v′ vec-tor essentially loads on only four
eigenvectors of the Σ matrix. These correlated eigenvectorsexplain
95% of the variations in v′ and they jointly capture three clusters
of sectors in theeconomy: 1) the heavy manufacturing sectors
including iron, steel, and machineries; 2) lightmanufacturing
sectors of consumer products including food and textiles; and 3)
the chemicalmanufacturing sectors. We show such a factor structure
emerges only when assessing theimpact of temporary shocks. In
contrast, the economy does not have a low-dimensional,factor
representation for permanent shocks, as the Domar weights have
significant loadingson over 150 eigenvectors. For simplicity, we
present our results assuming ρ = δ = 10% butthis choice is
immaterial for the results here.5
Table 1: Welfare elasticity to temporary sectoral shocks in the
U.S.
costs significantly down-weight the direct and initial rounds of
network effects, v′ may havea factor representation as long as |λk|
declines relatively fast in k.
5 Factor Structure of the U.S. Input-Output Table
We now turn to the 2012 U.S. input-output table published by the
U.S. Bureau of La-bor Statistics. We show that the high-dimensional
input-output table—171 by 171 sectorsunder broad categories of
agriculture, mining, manufacturing, and services4—has a
low-dimensional, 4-factor structure in terms of its susceptibility
to temporary shocks: the v′ vec-tor essentially loads on only four
eigenvectors of the Σ matrix. These correlated eigenvectorsexplain
95% of the variations in v′ and they jointly capture three clusters
of sectors in theeconomy: 1) the heavy manufacturing sectors
including iron, steel, and machineries; 2) lightmanufacturing
sectors of consumer products including food and textiles; and 3)
the chemicalmanufacturing sectors. We show such a factor structure
emerges only when assessing theimpact of temporary shocks. In
contrast, the economy does not have a low-dimensional,factor
representation for permanent shocks, as the Domar weights have
significant loadingson over 150 eigenvectors. For simplicity, we
present our results assuming ρ = δ = 10% butthis choice is
immaterial for the results here.5
Table 1: Welfare elasticity to temporary sectoral shocks in the
U.S.
10 sectors with the highest vi 10 sectors with the smallest
vi
Real estate Community and vocational rehabilitation services
Wholesale trade Gambling industries (except casino hotels)
Agencies, brokerages, and other insurance related activities
Other furniture related product manufacturing
Oil and gas extraction Personal care services
Basic chemical manufacturing Amusement parks and arcades
Management of companies and enterprises Grantmaking, giving
services, social advocacy organizations
Petroleum and coal products manufacturing Food and beverage
stores
Advertising, public relations, and related services Tobacco
manufacturing
Nonferrous metal (except aluminum) production & processing
Motor vehicle manufacturing
Motor vehicle parts manufacturing Other transportation equipment
manufacturing
413 sectors from the original 184-by-184 BLS input-output table
do not use or supply any intermediateinputs and therefore do not
interact with the rest of the network. These sectors are all in
services, includingoffices of dentists, individual family services,
home health care services, etc. We drop these sectors
whenperforming the eigendecomposition.
5In Online Appendix A, we show the 4-factor representation is
robust under alternative values of ρ andδ. In fact, the first 4
eigenvectors explain over 90% of the variation in v′ even in the
limit as ρδ →∞.
25
413 sectors from the original 184-by-184 BLS input-output table
do not use or supply any intermediateinputs and therefore do not
interact with the rest of the network. These sectors are all in
services, includingoffices of dentists, individual family services,
home health care services, etc. We drop these sectors
whenperforming the eigendecomposition.
5In Appendix A, we show the 4-factor representation is robust
under alternative values of ρ and δ. Infact, the first 4
eigenvectors explain over 90% of the variation in v′ even in the
limit as ρδ →∞.
24
-
Table 1 lists the top-10 most important and least important
sectors for the U.S. interms of v′, the welfare elasticity to
temporary sectoral shocks. As intuitions suggest, themost important
ones are large sectors that supply to many other producers. The
top-10list includes very large sectors such as real estate and
wholesale trade, whose sales-to-GDPratios add to 24%. The list also
includes much smaller but very upstream manufacturingsectors such
as chemical and metal sectors. On the right side of the table,
sectors with lowwelfare impact are those that are small and
downstream, including many service sectors.
Figure 3: Decay of eigen components
0 50 100 150
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0 50 100 150
0
0.5
1
1.5
2
2.5
3
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
We now describe the first empirical results of the paper: the
welfare elasticity to tempo-rary shocks in the U.S. can be
well-approximated in a low-dimensional sub-eigenspace.
In Figure 3, building on Proposition 5, the top panels (a)–(c)
respectively show theeigenvalues |λk|, their geometric sum |1/ (1−
λk)|, and the term
∣∣∣ λk(1−λk)(1+ρδ−λk)∣∣∣ for the U.S.
input-output table. The bottom panels (d)–(f) respectively show
|β′uk|, |β′uk/ (1− λk)|,and
∣∣∣β′uk λk(1−λk)(1+ρδ−λk)∣∣∣. In all panels, indices k are
arranged in decreasing order of the
absolute eigenvalues.
25
-
Panel (a) shows the decay of the eigenvalues of the input-output
matrix. The term|1/ (1− λk)| shown in panel (b) captures the
contribution of the k-th eigenvector to the geo-metric series in
the matrix (I −Σ)−1. The fact that |1/ (1− λk)| does not converge
to zero—it converges to 1 as |λk| → 0—implies that the matrix does
not have a low-dimensional repre-sentation, as even eigenvectors
after the 100th may be important.6 The term
∣∣∣ λk(1−λk)(1+ρδ−λk)∣∣∣
shown in panel (c), on the other hand, exhibit very rapid decay
towards zero; this implies
that the matrix (I −Σ)−1−(I − Σ
1+ρδ
)−1potentially has a low-dimensional representation.
Whether the Domar weight γ ′ = β′ (I −Σ)−1 or the welfare
elasticity to temporaryshocks v′ = 1
ρβ′[(I −Σ)−1 −
(I − Σ
1+ρδ
)−1]can be well-approximated in a low-dimensional
sub-eigenspace depends also on the loading of the consumption
vector β′ on each eigenvector.The loadings |β′uk| are shown in
panel (d), and the contribution of each k-th eigenvectorto the
Domar weight is shown in panel (e). As the figures shows, the
consumption vectorhas significant loadings on many eigenvectors,
and so does the Domar weight. For instance,both panels (d) and (e)
show spikes around the group of eigenvectors indexed around 75
to80—capturing sectors related to healthcare—and around 81 to
85—capturing sectors relatedto automobiles. These eigenvectors have
low eigenvalues, evidenced from panel (a), but theyare nevertheless
very important for the Domar weight because the consumer
expenditureshare β′ loads significantly on these eigenvectors,
meaning the consumption expenditureshares on healthcare and
automobiles are high. Jointly, the two “private hospitals”
and“motor vehicle manufacturing” sectors account for over 10% of
the final consumption share.This is even before accounting for
other related but smaller sectors such as “medical equip-ment and
supplies manufacturing”, “medical and diagnostic laboratories”,
“other ambulatoryhealth care services”, “motor vehicle body and
trailer manufacturing”, “motor vehicle partsmanufacturing”, and
“motor vehicle and parts dealers”.
We now turn to the analysis of the welfare elasticity to
temporary shocks, v′. Panel (f)stands in sharp contrast with panel
(e) and shows that only the initial few eigenvectors areimportant
in explaining variations in v′; that is, the impact of any
temporary TFP shockvector z̃ can be well-approximated by the
projection of the shock onto a low-dimensionalsub-eigenspace
spanned by the first few eigenvectors. As the discussion following
Proposition5 shows this is precisely due to the rapid decay of
∣∣∣ λk(1−λk)(1+ρδ−λk)∣∣∣ towards zero as shown in
panel (c). Even though β′ loads significantly onto some of the
high-indexed eigenvectors, thesectors underlying these eigenvectors
(e.g. hospitals and automobiles) are very downstream,meaning they
mostly supply directly to the final consumer and do not supply
strongly to otherintermediate sectors. Consequently, these
eigenvectors become unimportant in explaining the
6Certain entries in panel (b) are below one because some of the
eigenvalues are negative.
26
-
variation in v′.Figure 4 reproduces panels (d)–(f) of figure 3
by re-ordering the objects shown in each
panel according to declining absolute values (e.g., objects in
panel (d) is sorted in decliningorder of |β′uk| rather than
declining |λk|). Figure 4 confirms the message in figure 3
evenafter sorting: the importance of eigen component exhibits very
rapid decay in panel (f) areclose to zero after the few initial
components; by contrast, a large number of componentsremain
important in panels (d) and (e).
Figure 4: Decay of eigen components
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 50 100 150
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Howmany eigenvectors are needed to approximate v′? Let g′k ≡
λk(1−λk)(1+ρδ−λk)β′ukw′k
be the k-th eigen component of v′, and let v′(h) ≡ δ∑h
k=1 gk denote the partial sum of thefirst h eigen components.
Note that v′ = v′(N).
Figure 5 scatter plots v′(h) against v′ for h ≤ 6; the red line
in each panel is the 45-degree
line which indicates that v(h) is close to v. As the figure
shows, v′(4) approximates v′ very
well, and additional 5th and 6th eigen components do not seem to
significantly improve thefit.
27
-
Figure 5: Welfare impact from the initial eigenvectors (v′(h))
plotted against v′
0.0005
.001
.0015
0.0005
.001
.0015
0 .0005 .001 .0015 0 .0005 .001 .0015 0 .0005 .001 .0015
1 2 3
4 5 6
Table 2 shows the regression of v′(h) on v′ for h ∈ {1, . . . ,
6} and reports the slope
coefficients and adjusted R2. The results show that the first 3
eigenvectors capture 76% ofthe variation in v′; the first 4
eigenvectors capture 95% of the variation. That is, most ofthe
welfare impact of any sectoral shock z̃ can by explained by the
loading of the shock onthe first four eigenvectors.
Table 2: Regression of v′(h) on v′
h 1 2 3 4 5 6
slope 0.53 0.82 1.01 0.97 0.97 0.96
R2 0.39 0.58 0.76 0.95 0.96 0.94
Which sectors do the first four eigenvectors represent? Recall
that each eigenvectoruk represents a TFP shock profile, under which
the network effects decay at exponentiallyat rate λk and that the
cumulative welfare impact is −v′uk = − λk(1−λk)(1+ρδ−λk)β
′uk.
28
-
Figure 6: The first four eigenvectors of Σ
-.50
.5-.5
0.5
0 50 100 150 0 50 100 150
1 2
3 4
Metal products, foundries, forging
Manufacturing of consumer goods including food products,
textile, paper products, and furniture
Manufacturing of chemical products,
rubber, plastic Radio and television broadcasting
Agencies, brokerages, insurance
boiler tanks, machinery, electrical and
transportation equipment
Figure 6 visualizes the first four eigenvectors. The X-axis
represent the sectoral order-ing according to the BLS input-output
table, which roughly arranges broad sector groupsby agriculture,
food manufacturing, chemical products, metals, heavy manufacturing,
andservices. In the figure, we indicate the broad groups of sectors
that these eigenvectors repre-sent; Tables 4 and 5 provide more
detailed lists of sector names. Table 3 lists the first
foureigenvalues and the loading of the consumption vector on the
corresponding eigenvectors.Because input-output tables are not
symmetric, the eigenvectors are not orthogonal to eachother. In
fact, many eigenvectors are correlated, thereby picking up shocks
to the samegroups of sectors.
Table 3: Eigenvalues and the consumption loadings of the first
four eigenvector
k 1 2 3 4
λk 0.544 0.505 0.454 0.331
β′uk 0.035 0.034 0.043 0.027
29
-
The first eigenvector u1 represents shocks to the heavy
manufacturing sectors, includingmetal products, foundries, forging
and stamping, and as well as the production of boilertanks,
machinery, electrical and transportation equipment. This
eigenvector captures thevector of TFP shocks under which the
economic damage to GDP lasts the longest time afterTFP
recovers.
The second eigenvector u2 very strongly and negatively
correlates with the first, withPearson correlation coefficient of
−0.59 between u1 and u2. The eigenvector u2 activatesthree groups
of industries. First and most notably, u2 has large positive
entries for the twosectors relating to agencies, brokerages, and
insurance. Second, u2 has positive entries forthe manufacturing of
consumer goods including food, textile, paper products, and
furniture.Third, u2 has negative entries on the heavy manufacturing
industries, partly neutralizingthe shock profile from the first
eigenvector.
The third eigenvector u3 correlates positively with
u2—correlation coefficient 0.36—byhaving positive entries on the
manufacturing of consumer goods. In addition, u3 also
includessectors that manufacture chemicals, plastic, and rubber
products.
The fourth eigenvector has close-to-zero correlations with the
previous three eigenvectors.The new sector picked up by u4 is radio
and television broadcasting; in addition, u4 alsohas negative
entries on the manufacturing of chemicals, plastic, and rubber
products, partlyneutralizing the shock profiles represented by
u3.
Table 4: The 1st & 2nd eigenvector shock profiles: 10
largest entries by absolute value
[u1]i [u2]i
Nonferrous metal (except aluminum) production
and processing
0.439 Agencies, brokerages, and other insurance related
activities
0.574
Alumina and aluminum production and
processing
0.221 Insurance carriers 0.334
Other electrical equipment and component
manufacturing
0.213 Animal slaughtering and processing 0.128
Railroad rolling stock manufacturing 0.187 Dairy product
manufacturing 0.114
Motor vehicle manufacturing 0.178 Electrical equipment
manufacturing -0.097
Steel product manufacturing from purchased steel 0.178 Steel
product manufacturing from purchased steel -0.103
Forging and stamping 0.175 Forging and stamping -0.110
Boiler, tank, and shipping container
manufacturing
0.163 Alumina and aluminum production and
processing
-0.140
Iron and steel mills and ferroalloy manufacturing 0.159 Other
electrical equipment and component
manufacturing
-0.176
Motor vehicle parts manufacturing 0.153 Nonferrous metal (except
aluminum) production
and processing
-0.416
30
-
Table 5: The 3rd & 4th eigenvector shock profiles: 10
largest entries by absolute value
[u3]i [u4]i
Animal slaughtering and processing 0.314 Radio and television
broadcasting 0.628
Dairy product manufacturing 0.286 Animal slaughtering and
processing 0.362
Animal food manufacturing 0.211 Dairy product manufacturing
0.263
Resin, synthetic rubber, and artificial synthetic
fibers and filaments manufacturing
0.211 Alumina and aluminum production and
processing
0.180
Plastics product manufacturing 0.194 Railroad rolling stock
manufacturing 0.148
Textile mills and textile product mills 0.190 Paint, coating,
and adhesive manufacturing -0.129
Grain and oilseed milling 0.187 Rubber product manufacturing
-0.137
Sugar and confectionery product manufacturing 0.183 Textile
mills and textile product mills -0.186
Fruit and vegetable preserving and specialty food
manufacturing
0.179 Resin, synthetic rubber, and artificial synthetic
fibers and filaments manufacturing
-0.189
Animal production and aquaculture 0.168 Plastics product
manufacturing -0.196
Altogether, the eigenvectors u1 through u4 form a 4-dimensional
subspace of the 171-dimensional vector space in which the U.S.
input-output table lies. It may appear puzzling atfirst that the
sectors represented by these four eigenvectors do not seem to
coincide with thesectors with high welfare impacts as listed in
Table 1. There is no inconsistency: the welfareimpact of any
temporary TFP shock vector z̃ can be well-approximated by
projecting z̃ ontothis subspace, v′z̃ ≈ ∑4k=1 v′ukak, i.e.,
approximating z̃ with a linear combinations of u1through u4, with
coordinates ak = w′kz̃ obtained using the corresponding
left-eigenvectors.Table 6 shows the 4-dimensional coordinates for
shocking each of the 10 sectors with thehighest welfare impact
individually and no other sectors. As an example, the sector
“Agen-cies, brokerages, and other insurance related activities” has
a positive coordinate on u2,which picks up shocks to this sector
very strongly but also shocks to heavy manufacturingproducts
(negatively) and consumer goods (positively); see Figure 6. To
isolate the shock toagencies, brokerages and insurance, the sector
loads positively on u1 and negatively on u3to neutralize the other
sectors picked up by u2.
31
-
Table 6: Low dimensional representation of TFP shocks to
vulnerable sectors in the U.S.
10 sectors with the highest viLoadings on the first 4
eigenvectors
1st 2nd 3rd 4th
Real estate 0.29 0.10 0.62 1.05
Wholesale trade 0.44 0.02 0.70 0.39
Agencies, brokerages, and other insurance related activities
0.89 1.54 -1.50 -0.38
Oil and gas extraction 0.29 0.03 0.86 -0.64
Basic chemical manufacturing 0.46 0.06 1.75 -4.57
Management of companies and enterprises 0.17 0.03 0.37 0.20
Petroleum and coal products manufacturing 0.23 0.02 0.51
-0.15
Advertising, public relations, and related services 0.12 0.04
0.26 0.39
Nonferrous metal (except aluminum) production & processing
1.60 -0.20 -1.19 -0.43
Motor vehicle parts manufacturing 0.08 0.01 0.15 0.15
6 Leontief Targeting of Nazi Germany and Imperial Japan
Until now we have interpreted the vector v′ as the welfare
impact of slow recovery in sectoraloutput from temporary shocks to
TFP that recovers instantaneously. An equivalent inter-pretation of
v′ is the welfare impact if a sector’s production were to be halted
temporarilyand output were to be destroyed.
One of the first applications of input-output analysis developed
by Leontief was to estab-lish the priorities in targeting for
strategic bombing of the Axis powers, in particular NaziGermany and
Imperial Japan. Guglielmo (2008) describes that Leontief was a part
of theEnemy Objectives Unit (EOU), a group of economists tasked
with the analysis of targeting:
“The economists ... had a comparative advantage in answering the
... question[of How great is the impairment to the enemy’s war
efforts per unit of destruc-tion], which required familiarity with
the enemy’s industrial sector and the inputsrequired types of
output. This question could be quite complicated as a result ofthe
interconnectedness of the component sectors ... This insight, which
becameknown as input-output analysis would result in a Nobel Prize
in economics.”
One important concept was that of depth, a measure of how long
it took for damage to havean impact on enemy capacity on the
battlefield. The final products such as tanks had less
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depth compared to the intermediate products.The EOU memoranda
compiled by Rostow (1981) discuss the criteria for target
selection
by the EOU economists. For instance, Salant (1942) wrote in one
memorandum:
“it is better to attack a factory the loss of whose output will
have widespreadeffect in causing stoppages elsewhere than one which
is a relatively isolated unitin the industrial system.”
The E.O.U. Special Report No.9 (1943) also stated
“the most attractive target groups, for general attack on war
production are...in the range of components: bearings, the Bosch
line, tires, and the other familiaritems. It is clear that a time
interval will elapse....”
Harrison (2020) summarizes the strategy of economic warfare by
the Allies as indirectlyattacking the enemy through its supply
chain. Bollard (2020) further argued that
“Leontief ’s input-output provided an economic mapping that the
generals couldreadily understand.”
There are also several criticisms of this input-output strategy.
First, the United StatesStrategic Bombing Survey conducted after
the war to assess the effectiveness of strategicbombing concluded
that bombing had a limited impact on the Nazi economy
(Guglielmo(2008)). Similarly, the input-output models assumed the
fixed coefficients while in reality theNazi economy was able to
substitute to, perhaps less efficient, but still workable
alternatives(Olson (1962), Harrison (2020)).
While, of course, there were many other reasons for target
selection such as politicaland military aims, our model is a modern
version of the analysis of the EOU economistsfighting against the
Axis. In our setup, the welfare impact vi can be seen as the
aggregateeconomic impact of damaging sector i in Nazi Germany and
Imperial Japan. In what follows,we estimate the economic impact of
shocking each sector for these two countries before theWorld War
II. More broadly, we use this section to also illustrate a variety
of other featuresof our model such as the cross-sectoral impact of
shocks over several time horizons.
Specifically, we digitize the 40-by-40 industries input-output
table of Germany in 1936from Fremdling and Staeglin (01 Nov. 2014),
and we translate and digitize the 23-by-23 industries input-output
table of pre-war Japan in 1935 from Nishikawa and Koshihara(1981).7
Of course, these data sources were not available to EOU economists
at the time and
7Because the input-output table of Japan in 1935 is not
available from digital sources, we provide ourdigitized version in
Appendix B Table 11.
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they almost surely did not fully represent the mobilized war
time economy, but it is usefulto analyze them as they are the best
available current sources of information.
First, we provide the list of sectors to which temporary shocks
generate the largest impact.Second, we show, for the purpose of
finding vulnerability to temporary shocks, both of
theseinput-output tables also exhibit low-dimensional
representations: the first three eigenvectorsexplain 92% of the
variation in v′ for Imperial Japan and 85% for Nazi Germany.
Third,we provide an interpretation of the first three eigenvectors
for both economies. Fourth,we demonstrate the over-time impact of
shocks to each sector on every other sector of theeconomy, and we
show shocks to the metal sectors tend to have lasting damage across
bothfor the pre-WWII Germany and Japan.
Table 7 shows the top-5 most important sectors in terms of v′,
the welfare elasticity totemporary shocks, for Germany in 1936
(left panel) and Japan in 1935 (right panel). Ironand steel
products, or metals in general, are important for both economies
because they areupstream and because shocks to these sectors
destroy many network linkages that could takea long time to
recover; we provide further evidence below. Textiles and
agriculture are alsoimportant for both economies because these
sectors represent a significant fraction of GDP.
Table 7: Welfare elasticity to temporary sectoral shocks in
pre-WWII Germany and Japan
5 sectors with the highest vi for Germany in 1936 5 sectors with
the highest vi for Japan in 1935
Basic iron and steel products Agriculture, forestry
Transport and communication Metals
Other services Chemicals
Textiles Textile/personal goods
Agriculture Commerce
Recall that v′(h) is the h-dimensional approximation of v′ based
on the first h eigen com-
ponents. Table 8 shows the regression of v′(h) on v′ for h ≤ 4
and reports the slope coefficients
and adjusted R2. The results show that the first 3 eigenvectors
capture 85% and 92% of thevariation respectively for the pre-WWII
Germany and Japan, implying that, similar to themodern U.S.
economy, most of the welfare impact of any sectoral shock z̃ in
these pre-WWIIeconomies can by explained by the loading of the
shock on the first three eigenvectors in theseeconomies. As we have
explained, such low-dimensional representation does not exist for
theDomar weight, as the representation is possible only because v′
significantly up-weights theimportance of eigen components with
large eigenvalues.
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Table 8: Regression of v′(h) on v′ for pre-WWII Germany and
Japan
Germany Japan
h 1 2 3 4 1 2 3 4
slope 0.87 0.95 1.02 10.4 0.64 0.65 0.91 0.92R2 0.67 0.66 0.85
0.87 0.49 0.65 0.92 0.92
Table 9 describes the first three eigenvectors for these
economies by listing the 5 largestsectoral entries by absolute
value for each eigenvector. For Germany, the first two
eigen-vectors are highly correlated and represent the iron, steel,
vehicles and aerospace industries.The first eigenvector also loads
strongly on the spirits industry but the loading is negatedby the
second eigenvector. The third eigenvector loads on textile,
clothing, fuel, and non-ferrous metals. For Japan, because the
input-output table only contains 23 sectors, we haveto examine the
industry structures with limited resolution. Nevertheless, the
first eigenvec-tor loads strongly on metals, machinery, and
construction; the first three eigenvectors alsojointly loads on
textiles, office supplies, printing and publishing, and leather and
rubberproducts.
Proposition 2 characterizes the entire path of the sectoral
output vector ln q (t) as afunction of the initial shock vector z̃
and time. One can also apply the proposition tocompute the
half-lives of shocks, as summarized in the proposition below.
Proposition 6. The half-life ti1/2 of temporary TFP shock vector
−z̃ on sector i’s output isthe solution to
e′iΣ (I −Σ)−1(
1
2I − e−δ−1(I−Σ)tc1/2
)z̃ = 0.
The half-life tc1/2 of temporary TFP shock vector −z̃ on
aggregate consumption is the solutionto
β′Σ (I −Σ)−1(
1
2I − e−δ−1(I−Σ)tc1/2
)z̃ = 0.